Attention surfers! You may have been routed to this page directly from the WWW. This page is part of Marco Avellaneda's NYU website . I recommend that you visit the main page for updated contents, and that you visit the Courant Finance Server , which is more up-to-date.

Topics in Probability: The mathematics of financial risk-management

Instructor: Marco Avellaneda

Courant Institute of Mathematical Sciences

Spring, 1996

(Initial blurb (1996)) This course is divided into four parts. First, we will cover fundamentals on Ito Calculus and Arbitrage Pricing Theory, emphasizing the use of trinomial trees, or finite-difference schemes, for pricing derivative securities (options, forwards, callable bonds, etc.) Next, we will move into the area of transaction costs due to liquidity constraints. We will analyze how low liquidity in trading the underlying asset affects implied volatility and dynamic hedging. Third, we will consider different ways of of modeling markets with uncertain volatility. We will cover stochastic volatility models, auto-regressive models as well as recently derived nonlinear pricing models. We will emphasize, in particular, worst-case scenario risk-management techniques, and managing volatility risk with options. Finally, we will cover more practical risk-management techniques and guidelines such as the ``Value at Risk'' outlined in the J.P. Morgan technical document on measuring financial risk.

Mathematics of Finance II

(Spring 1998) Mathematics of Finance II borrows some of the same basic material from the original Risk-management course but emphasizes mostly (i) continous time finance and (ii) modeling the term structure of interest rates. We do a little bit of ``taxonomy'' of models as well as some conceptual thinking about the significance and applications of term-structure models. Topics tend to change from one year to the next.

Lecture Notes

1. Syllabus.

2. Brownian Motion and Ito Calculus.

3. Ito processes, continuous-time martingales and Girsanov's Theorem (revised)

4. Continuous-time finance: an introduction

5. Valuation of derivative securities

6. Uncertain Volatility Model & worst-case scenario pricing
.ps file of transparencies. For more complete notes, look at the first two papers
in ``Recent papers in Mathematical Finance''.

7. Trinomial trees and finite-difference schemes

INTEREST RATE DERIVATIVES (Spring 1998, very rough drafts)

8. Basic term structure concepts
9. Heath-Jarrow-Morton thm and forward rate correlations
10. Affine term structure models

NEW PROBLEMS (Spring 1998)

#1 Brownian exit time from a strip
(2nd corrected version: thanks to those that pointed out earlier mistakes! :-) )

#2 Gauss-Markov processes

#3 Term-structure Modeling

Old Homework Assignments

Homework #1 (revised)

Extra problem: digital barrier and volalility term-structure

Homework # 2